nLab additive functor



Enriched category theory

Additive and abelian categories

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homological algebra

(also nonabelian homological algebra)



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diagram chasing

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A functor F:π’œβ†’β„¬F: \mathcal{A} \to \mathcal{B} between additive categories is itself called additive if it preserves finite biproducts.

That is,

  1. FF maps a zero object to a zero object, F(0)≃0βˆˆβ„¬F(0) \simeq 0 \in \mathcal{B};

  2. given any two objects x,yβˆˆπ’œx, y \in \mathcal{A}, there is an isomorphism F(xβŠ•y)β‰…F(x)βŠ•F(y)F(x \oplus y) \cong F(x) \oplus F(y), and this respects the inclusion and projection maps of the direct sum:

x y i xβ†˜ ↙ i y xβŠ•y p x↙ β†˜ p y x y↦FF(x) F(y) i F(x)β†˜ ↙ i F(y) F(xβŠ•y)β‰…F(x)βŠ•F(y) p F(X)↙ β†˜ p F(y) F(x) F(y) \array { x & & & & y \\ & {}_{\mathllap{i_x}}\searrow & & \swarrow_{\mathrlap{i_y}} \\ & & x \oplus y \\ & {}^{\mathllap{p_x}}\swarrow & & \searrow^{\mathrlap{p_y}} \\ x & & & & y } \quad\quad\stackrel{F}{\mapsto}\quad\quad \array { F(x) & & & & F(y) \\ & {}_{\mathllap{i_{F(x)}}}\searrow & & \swarrow_{\mathrlap{i_{F(y)}}} \\ & & F(x \oplus y) \cong F(x) \oplus F(y) \\ & {}^{\mathllap{p_{F(X)}}}\swarrow & & \searrow^{\mathrlap{p_{F(y)}}} \\ F(x) & & & & F(y) }

In practice, functors between additive categories are generally assumed to be additive.


Each of the following conditions is sufficient for guaranteeing that a functor π’œβ†’β„¬\mathcal{A} \to \mathcal{B} preserves biproducts (where π’œ\mathcal{A} and ℬ\mathcal{B} are categories with a zero object):

  • The functor preserves finite products (for instance, because it’s a right adjoint) and any product in ℬ\mathcal{B} is a biproduct.
  • The functor preserves finite coproducts (for instance, because it’s a left adjoint) and any coproduct in ℬ\mathcal{B} is a biproduct.
  • The functor preserves finite products and coproducts.



The hom-functor Hom(βˆ’,βˆ’):π’œ opΓ—π’œβ†’AbHom(-,-) \colon \mathcal{A}^{op}\times \mathcal{A} \to Ab is additive in both arguments separately (using the nature of biproducts and that hom-functors preserve limits in each variable separately).


For π’œ=R\mathcal{A} = RMod and Nβˆˆπ’œN \in \mathcal{A}, the functor that forms tensor product of modules (βˆ’)βŠ—N:π’œβ†’π’œ(-)\otimes N \colon \mathcal{A} \to \mathcal{A}.

In fact these examples are generic, see prop. below.


Every solid abelian group is by definition an additive functor.


Relation to AbAb-enriched functors

An additive category canonically carries the structure of an Ab-enriched category where the AbAb-enrichment structure is induced from the biproducts as described at biproduct.


With respect to the canonical Ab-enriched category-structure on additive categories π’œ\mathcal{A}, ℬ\mathcal{B}, additive functors F:π’œβ†’β„¬F : \mathcal{A} \to \mathcal{B} are equivalently Ab-enriched functors.


An AbAb-enriched functor preserves all finite biproducts that exist, since finite biproducts in Ab-enriched categories are Cauchy colimits.

Characterization of right exact additive functors

Let R,Rβ€²R, R' be rings.

The following is the Eilenberg-Watts theorem. See there for more.


If an additive functor F:RF : RMod β†’Rβ€²\to R'Mod is a right exact functor, then there exists an Rβ€²R'-RR-bimodule BB and a natural isomorphism

F≃BβŠ— R(βˆ’) F \simeq B \otimes_R (-)

with the functor that forms the tensor product with BB.

This is (Watts, theorem 1),


  • Charles Watts, Intrinsic characterizations of some additive functors, Proceedings of the American Mathematical Society, 11 1 (1960) 5-8 (1959) [jstor:2032707]

Last revised on April 15, 2023 at 09:59:22. See the history of this page for a list of all contributions to it.